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Linear-cosmology observables, such as the Cosmic Microwave Background (CMB),or the large-scale distribution of matter, have long been used as clean probesof dark matter (DM) interactions with baryons. It is standard to model the DMas an ideal fluid with a thermal Maxwell-Boltzmann (MB) velocity distribution,in order to compute the heat and momentum-exchange rates relevant to theseprobes. This approximation only applies in the limit where DM self-interactionsare frequent enough to efficiently redistribute DM velocities. It does notaccurately describe weakly self-interacting particles, whose velocitydistribution unavoidably departs from MB once they decouple from baryons. Thisarticle lays out a new formalism required to accurately model DM-baryonscattering, even when DM self-interactions are negligible. The ideal fluidequations are replaced by the collisional Boltzmann equation for the DMphase-space distribution. The collision operator is approximated by aFokker-Planck operator, constructed to recover the exact heat and momentumexchange rates, and allowing for an efficient numerical implementation.Numerical solutions to the background evolution are presented, which show thatthe MB approximation can over-estimate the heat-exchange rate by factors of ~2-3, especially for light DM particles. A Boltzmann-Fokker-Planck hierarchy forperturbations is derived. This new formalism allows to explore a wider range ofDM models, and will be especially relevant for upcoming ultra-high-sensitivityCMB probes.

Boltzmann-Fokker-Planck formalism for dark-matter--baryon scattering